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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Hilbert–Poincaré series ： ウィキペディア英語版 Hilbert–Poincaré series In mathematics, and in particular in the field of algebra, a Hilbert–Poincaré series (also known under the name Hilbert series), named after David Hilbert and Henri Poincaré, is an adaptation of the notion of dimension to the context of graded algebraic structures (where the dimension of the entire structure is often infinite). It is a formal power series in one indeterminate, say ''t'', where the coefficient of ''t''''n'' gives the dimension (or rank) of the sub-structure of elements homogeneous of degree ''n''. It is closely related to the Hilbert polynomial in cases when the latter exists; however, the Hilbert–Poincaré series describes the rank in every degree, while the Hilbert polynomial describes it only in all but finitely many degrees, and therefore provides less information. In particular the Hilbert–Poincaré series cannot be deduced from the Hilbert polynomial even if the latter exists. In good cases, the Hilbert–Poincaré series can be expressed as a rational function of its argument ''t''. == Definition == Let ''K'' be a field, and let $V=\backslash textstyle\backslash bigoplus\_\}\backslash dim\_K(V\_i)t^i.$ A similar definition can be given for an N-graded ''R''-module over any commutative ring ''R'' in which each submodule of elements homogeneous of a fixed degree ''n'' is free of finite rank; it suffices to replace the dimension by the rank. Often the graded vector space or module of which the Hilbert–Poincaré series is considered has additional structure, for instance that of a ring, but the Hilbert–Poincaré series is independent of the multiplicative or other structure. Example: Since there are $\backslash binom$ monomials of degree ''k'' in variables $X\_0,\; \backslash dots,\; X\_n$ (by induction, say), it follows immediately that the Hilbert–Poincaré series of ''K''() is $(1-t)^$ 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Hilbert–Poincaré series」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.